Properties of the Grothendieck group

The Grothendieck construction has the following properties.

  1. Universal property
    If is an abelian group, and if is an additive map, then there is one, and only one group homomorphism making the diagram commute.
  2. Functoriality
    To every additive map between semigroups and there is precisely one group homomorphism which makes the diagram commute.
  3. Let be elements in . Then if and only if for some .
  4. The Grothendieck map is injective if and only if has the cancellation property.
  5. Let be an abelian group, and let be a non-empty subset of . If is closed under addition, then is an abelian semigroup with the cancellation property, is isomorphic to the subgroup generated by , and .